## Modern statistics for spatial point processes ^{∗}

### June 21, 2006

Jesper Møller and Rasmus P. Waagepetersen

Department of Mathematical Sciences, Aalborg University

Abstract: We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs, and Cox process models, diagnostic tools and model checking, Markov chain Monte Carlo algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.

Keywords: Bayesian inference, conditional intensity, Cox process, Gibbs point process, Markov chain Monte Carlo, maximum likelihood, perfect simulation, Poisson process, residuals, simulation free estimation, summary statistics.

### 1 Introduction

Spatial point pattern data occur frequently in a wide variety of scientific disci- plines, including seismology, ecology, forestry, geography, spatial epidemiology, and material science, see e.g. Stoyan & Stoyan (1998), Kerscher (2000), Boots, Okabe & Thomas (2003), Diggle (2003), and Ballani (2006). The classical spa- tial point process textbooks (Ripley, 1981, 1988; Diggle, 1983; Stoyan, Kendall

& Mecke, 1995; Stoyan & Stoyan, 1995) usually deal with relatively small point

∗Prepared for presentation as an special invited talk at the 21st Nordic Conference on Mathematical Statistics, June 11 - 15, 2006, and for submission to the Scandinavian Journal of Statistics.

patterns, where the assumption of stationarity is central and non-parametric methods based on summary statistics play a major role. In recent years, fast computers and advances in computational statistics, particularly Markov chain Monte Carlo (MCMC) methods, have had a major impact on the develop- ment of statistics for spatial point processes. The focus has now changed to likelihood-based inference for flexible parametric models, often depending on covariates, and liberated from restrictive assumptions of stationarity. In short,

‘Modern statistics for spatial point processes’, where recent textbooks include Van Lieshout (2000), Diggle (2003), Møller & Waagepetersen (2003b), and Bad- deley, Gregori, Mateu, Stoica & Stoyan (2006).

Much of the literature on spatial point processes is fairly technical with ex- tensive use of measure theoretical terminology and statistical physics parlance.

This has made the theory seem rather difficult. Moreover, in connection with likelihood-based inference, many statisticians may be unfamiliar with the con- cept of defining a density with respect to a Poisson process. It is our intention in Sections 3–9 to give a concise and non-technical introduction to the modern theory, making analogies with generalized linear models and random effect mod- els, and illustrating the theory with various examples of applications introduced in Section 2. In particular, we discuss Poisson, Gibbs, and Cox process mod- els, diagnostic tools and model checking, MCMC algorithms and computational methods for likelihood-based inference, and quick non-likelihood approaches to inference. Section 10 summarizes the current state of spatial point process the- ory and discusses directions for future research.

For definiteness, we mostly work with point processes defined in the plane
R^{2}, but most ideas easily extend to the general case of R^{d} or more abstract
spaces. For ease of exposition, no measure theoretical details are given; see
instead Møller & Waagepetersen (2003b) and the references therein. The com-
putations for the data examples were done using theRpackagespatstat(Bad-
deley & Turner, 2005, 2006) or our own programmes inCandR, where the code
is available at www.math.aau.dk/~rw/sppcode. Since we shall often refer to
our own monograph, please notice the comments and corrections to Møller &

Waagepetersen (2003b) atwww.math.aau.dk/~jm.

### 2 Data examples

The following four examples of spatial point pattern data are from plant and animal ecology, and are considered for illustrative purposes in subsequent sec- tions. In each example, theobservation windowrefers to the area where points of the pattern can possibly be observed, i.e. when the point pattern is viewed as a realization of a spatial point process (Section 3.1). Absence of points in a region, where they could potentially occur, is a source of information comple- mentary to the data on where points actually did occur. The specification of the observation window is therefore an integral part of a spatial point pattern data set.

Figure 1 shows positions of 55 minke whales (Balaneoptera acutorostrata) observed in a part of the North Atlantic near Spitzbergen. The whales are observed visually from a ship sailing along predetermined so-called transect lines. The point pattern can be thought of as an incomplete observation of all the whale positions, since it is only possible to observe whales within the vicinity of the ship. Moreover, whales within sighting distance may fail to be observed due to bad weather conditions or if they are diving. The probability of observing a whale is a decreasing function of the distance from the whale to the ship and is effectively zero for distances larger than 2 km. The observation window is therefore a union of narrow strips of width 4 km around the transect lines. The data in Figure 1 do not reflect the fact that the whales move and that the whales are observed at different points in time. However, the observations from different transect lines may be considered approximately independent due to the large spatial separation between the transect lines. More details on the data set and analysis of line transect data can be found in Skaug, Øien, Schweder & Bøthun (2004), Waagepetersen & Schweder (2006), and Buckland, Anderson, Burnham, Laake, Borchers, and Thomas (2004). The objective is to estimate the abundance of the whales, or equivalently the whale intensity.

The whales tend to cluster around locations of high prey intensity, and a point process model for all whale positions (including those not observed) should take this into account. The point process model used in Waagepetersen & Schweder (2006) is described in Example 4.2.

In studies of biodiversity of tropical rain forests, it is of interest to study whether the spatial patterns of the many different tree species can be related to spatial variations in environmental variables concerning topography and soil properties. Figure 2 shows positions of 3605Beilschmiedia pendula Lauraceae

Figure 1: Observed whales along transect lines. The enclosing rectangle is of dimensions 263 km by 116 km.

trees in the tropical rain forest of Barro Colorado Island. This data set is a part of a much larger data set containing positions of hundreds of thousands of trees belonging to thousands of species, see Hubbell & Foster (1983), Condit, Hubbell

& Foster (1996), and Condit (1998). In addition to the tree positions, covariate information on altitude and norm of altitude gradient is available, see Figure 3.

Phrased in point process terminology, the question is whether the intensity of Beilschmiediatrees may be viewed as a spatially varying function of the covari- ates. In the study of this question, it is, as for the whales, important to take into account clustering, which in the present case may be due to tree reproduction by seed dispersal and possibly unobserved covariates. Different point process models for the tree positions are considered in Examples 4.1 and 4.3.

Figure 2: Locations ofBeilschmiedia pendula Lauraceaetrees observed in a 1000 m by 500 m rectangular window.

Another pertinent question in plant ecology is how trees interact due to competition. Figure 4 shows positions and stem diameters of 134 Norwegian

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0 200 400 600 800 1000 1200

−1000100200300400500600 00.050.150.250.35

Figure 3: Altitude (upper plot) in meter and norm of altitude gradient (lower plot).

spruces. This data set was collected in Tharandter Forest, Germany, by the forester G. Klier and was first analyzed using point process methods by Fiksel (1984). The data set is an example of a marked point pattern, with points given by the tree locations and marks by the stem diameters. The discs in Figure 4 are of radii five times the stem diameters and may be thought of as

‘influence zones’ of the trees, see Penttinen, Stoyan & Henttonen (1992) and Goulard, S¨arkk¨a & Grabarnik (1996). The regularity in the point pattern is to a large extent due to forest management. From an ecological point of view it is of interest to study how neighbouring trees interact, i.e. when their influence zones overlap. It is then natural to model the conditional intensity, which roughly speaking determines the probability of observing a tree at a given location and of given stem diameter conditional on the neighbouring trees. In Example 5.1, we consider a simple model where the conditional intensity depends on the

amount of overlap between the influence zones of a tree and its neighbouring trees.

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Figure 4: Norwegian spruces observed in a rectangular 56 m by 38 m window.

The radii of the discs equal 5 times the stem diameters.

Our last data set is an example of a multitype point pattern, with two types of points specifying the positions of nests for two types of ants, Messor Was- manni and Cataglyphis Bicolor, see Figure 5 and Harkness & Isham (1983).

Note the rather atypical shape of the observation window. The interaction be- tween the two types of ants is of main interest for this data set. Biological knowledge suggests that theMessorants are not influenced by presence or ab- sence ofCataglyphisants when choosing sites for their nests. The catagplyphis ants, on the other hand, feed on deadMessorsand hence the positions ofMes- sornests might affect the choice of sites forCataglyphisnests. H¨ogmander and S¨arkk¨a (1999) therefore specify a hierarchical model: first a model for the condi- tional intensity of aMessornest at a particular location given the neighbouring Messor nests, and second a conditional intensity for a Cataglyphis nest given the neighbouringCataglyphisnestsandthe neighbouringMessornests. Further details are given in Example 5.2.

These examples illustrate many important features of interest for spatial point process analysis: clusteringdue to e.g. seed dispersal or unobserved vari- ation in prey intensity (as for the tropical rain forest trees and the whales), inhomogeneity e.g. caused by a thinning mechanism or covariates (as for the

Figure 5: Locations of nests for Messor (triangles) and Cataglyphis (circles) ants. Enclosing rectangle for observation window is 829 ft by 766 ft.

whales and tropical rain forest trees), and interaction between points, where the interaction possibly depends on marks associated with the points (as for the Norwegian spruces). The examples also illustrate different types of observation windows.

### 3 Preliminaries

### 3.1 Spatial point processes

In the simplest case, a spatial point process X is a finite random subset of a
given bounded regionS ⊂R^{2}, and a realization of such a process is aspatial
point patternx={x1, . . . , xn}ofn≥0 points contained inS. We say that the
point process is defined onS, and we writex=∅for the empty point pattern.

The number of points,n(X), is a random variable, and an equivalent approach is to specify the distribution of the variablesN(B) =n(XB) for subsetsB⊆S, whereXB=X∩B.

If it is not known on which region the point process is defined, or if the pro- cess extends over a very large region, or if certain invariance assumptions such

as stationarity are imposed, then it may be appropriate to consider an infinite
point process onR^{2}. We define aspatial point processXonR^{2}as a locally finite
random subset of R^{2}, i.e. N(B) is a finite random variable whenever B ⊂R^{2}
is a bounded region. We say that X is stationary respective isotropic if its
distribution is invariant under translations inR^{2}respective rotations about the
origin inR^{2}. Stationarity and isotropy may be reasonable assumptions for point
processes observed within a homogeneous environment. These assumptions ap-
peared commonly in the older point process literature, where typically rather
small study regions were considered. We shall abandon these assumptions when
spatial covariate information is available.

In most applications, the observation windowW (see Section 2) is strictly contained in the regionS, where the point process is defined. Since X\W is unobserved, we face a missing data problem, which in the spatial point process literature is referred to as a problem ofedge effects.

### 3.2 Moments

The mean structure of the count variablesN(B), B ⊆ R^{2}, is summarized by
themoment measure

µ(B) = EN(B), B ⊆R^{2}. (1)

In practice the mean structure is modelled in terms of a non-negativeintensity functionρ, i.e.

µ(B) = Z

B

ρ(u) du

where we may interpretρ(u) duas the probability that precisely one point falls in an infinitesimally small region containing the locationuand of area du.

The covariance structure of the count variables is most conveniently given
in terms of thesecond order factorial moment measureµ^{(2)}. This is defined by

µ^{(2)}(A) = E

6=

X

u,v∈X

1[(u, v)∈A], A⊆R^{2}×R^{2}, (2)

where 6= over the summation sign means that the sum runs over all pairwise
different pointsu, vinX, and1[·] is the indicator function. For bounded regions
B⊆R^{2} andC⊆R^{2},

Cov[N(B), N(C)] =µ^{(2)}(B×C) +µ(B∩C)−µ(B)µ(C).

For many important model classes,µ^{(2)} is given in terms of an explicitly known
second order product densityρ^{(2)},

µ^{(2)}(A) =
Z

1[(u, v)∈A]ρ^{(2)}(u, v) dudv

whereρ^{(2)}(u, v)dudvmay be interpreted as the probability of observing a point
in each of two regions of infinitesimally small areas duand dv and containing
uand v. More generally, for integers n ≥1, the nth order factorial moment
measureµ^{(n)} is defined by

µ^{(n)}(A) = E

6=

X

u1,...,un∈X

1[(u1, . . . , un)∈A], A⊆R^{2n}, (3)

with correspondingnth order product densityρ^{(n)}. From (3) we obtain Camp-
bell’s theorem

E

6=

X

u1,...,un∈X

h(u1, . . . , un) = Z

h(u1, . . . , un)ρ^{(n)}(u1, . . . , un) du1,· · ·dun (4)

for non-negative functionsh. The nth order moment measure is given by the right hand side of (3) without6=. The reason for preferring the factorial moment measures are the nicer expressions for the product densities, cf. (6) and (16).

In order to characterize the tendency of points to attract or repel each other, while adjusting for the effect of a large or small intensity function, it is useful to consider thepair correlation function

g(u, v) =ρ^{(2)}(u, v)/(ρ(u)ρ(v)) (5)
(providedρ(u)>0 andρ(v)>0). If points appear independently of each other,
ρ^{(2)}(u, v) = ρ(u)ρ(v) and g(u, v) = 1 (see also (6)). When g(u, v) >1 we in-
terpret this as attraction between points of the process at locations u and v,
while ifg(u, v)<1 we haverepulsionat the two locations. Translation invari-
anceg(u, v) =g(u−v) ofg implies thatXissecond order intensity reweighted
stationary(Baddeley, Møller & Waagepetersen, 2000 and Section 6.2.1), and in
applications it is often assumed thatg(u, v) =g(ku−vk) depends only on the
distanceku−vk. Notice that very different point process models can share the
same g function (Baddeley & Silverman, 1984, Baddeley et al., 2000, Diggle,
2003 (Section 5.8.3)).

Suppose π(u) ∈ [0,1], u ∈ R^{2}, are given numbers. An independent π-
thinning of X is obtained by independently retaining each point uin X with
probabilityπ(u). It follows easily from (4) thatπ(u1)· · ·π(un)ρ^{(n)}(u1, . . . , un)
is thenth order product density of the thinned process. In particular,π(u)ρ(u)
is the intensity function of the thinned process, whileg is the same for the two
processes.

### 3.3 Marked point processes

In addition to each point u in a spatial point process X, we may have an
associated random variable mu called a mark. The mark often carries some
information about the point, for example the radius of a disc as in Figure 4,
the type of ants as in Figure 5, or another point process (e.g. the clusters in a
shot noise Cox process, see Section 4.2.2). The processΦ={(u, mu) :u∈X}
is called amarked point process, see Stoyan & Stoyan (1995), Schlather (2001),
and Møller & Waagepetersen (2003b). For the models presented later in this
paper, the marked point process model of discs in Figure 4 will be viewed as a
point process inR^{2}×(0,∞), and the bivariate point process model of ants nests
in Figure 5 will be specified by a hierarchical model so that no methodology
specific to marked point processes is needed.

### 3.4 Generic notation

Unless otherwise stated,

Xdenotes a generic spatial point process defined on a regionS⊆R^{2};
W ⊆S is a bounded observation window;

x={x1, . . . , xn} is either a generic finite point configuration or a realization ofXW (the meaning ofxwill always be clear from the context);

z(u) = (z1(u), . . . , zk(u)) is a vector of covariates depending on locationsu∈ S such as spatially varying environmental variables, known functions of the spatial coordinates themselves or distances to known environmental features, cf. Berman & Turner (1992) and Rathbun (1996);

β= (β1, . . . , βk) is a corresponding regression parameter;

θis the vector of all parameters (including β) in a given parametric model.

### 4 Modelling the intensity function

This section discusses spatial point process models specified by a determinis- tic or random intensity function by analogy with generalized linear models and random effects models. Particularly, two important model classes, namely Pois- son and Cox/cluster point processes are introduced. Roughly speaking, the two classes provide models for no interaction and aggregated point patterns, respectively.

### 4.1 The Poisson process

APoisson processX defined onS and with intensity measure µand intensity functionρsatisfies for any bounded regionB⊆S withµ(B)>0,

(i) N(B) is Poisson distributed with meanµ(B),

(ii) conditional onN(B), the points inXB are i.i.d. with density proportional toρ(u),u∈B.

Poisson processes are studied in detail in Kingman (1993). They play a funda- mental role as a reference process for exploratory and diagnostic tools and when more advanced spatial point process models are constructed.

Ifρ(u) is constant for allu∈S, we say that the Poisson process ishomoge- neous. Realizations of the process may appear to be rather chaotic with large empty space and close pairs of points, even when the process is homogeneous.

The Poisson process is a model for ‘no interaction’ or ‘complete spatial ran- domness’, since XA andXB are independent whenever A, B ⊂S are disjoint.

Moreover,

ρ^{(n)}(u1, . . . , un) =ρ(u1)· · ·ρ(un), g≡1, (6)
reflecting the lack of interaction. Stationarity means thatρ(u) is constant, and
implies isotropy ofX. Note that another Poisson process results if we make an
independent thinning of a Poisson process.

Typically, a log linear model of the intensity function is considered (Cox, 1972),

logρ(u) =z(u)β^{T}. (7)

The independence properties of a Poisson process are usually not realistic for real data. Despite of this the Poisson process has enjoyed much popularity due to its mathematical tractability.

### 4.2 Cox processes

One natural extension of the Poisson process is aCox processXdriven by a non-
negative process Λ = (Λ(u))_{u∈S}, such that conditional on Λ, X is a Poisson
process with intensity function Λ (Cox, 1955; Mat´ern, 1971; Grandell, 1976;

Daley & Vere-Jones, 2003).

Three points of statistical importance should be noticed. First, though Λ may be modelling a random environmental heterogeneity, X is stationary if Λ is stationary. Second, we cannot distinguish the Cox process X from its corresponding Poisson processX|Λwhen only one realization ofXW is available, cf. Møller & Waagepetersen (2003b, Section 5.1). Third, the likelihood is in general unknown, while product densities may be tractable. The consequences of the latter point are discussed in Sections 7 and 8.

4.2.1 Log Gaussian Cox processes

In analogy with random effect models, as an extension of the log linear model (7), take

log Λ(u) =z(u)β^{T}+ Ψ(u) (8)

where Ψ = (Ψ(u))u∈S is a zero-mean Gaussian process. Then we call X a log Gaussian Cox process (Møller, Syversveen & Waagepetersen, 1998). The covariance functionc(u, v) = Cov[Ψ(u),Ψ(v)] typically depends on some lower- dimensional parameter, see e.g. Example 4.1 below. To ensure local integrability of Λ(u), the covariance function has to satisfy certain mild conditions, which are satisfied for models used in practice.

The product densities are particularly tractable. The intensity function
logρ(u) =z(u)β^{T}+c(u, u)/2 (9)
is log linear,g andcare in a one-to-one correspondence as

g(u, v) = exp(c(u, v))

and higher-order product densities are nicely expressed in terms of ρ and g (Mølleret al., 1998). Another advantageous property is that we have no problem with edge effects, since XW is specified by the Gaussian process restricted to W.

Example 4.1. (Log Gaussian Cox process model for tropical rain forest trees)

For the tropical rain forest trees in Figure 2, we consider in Example 7.5 infer-
ence for a log Gaussian Cox process withz(u) = (1, z2(u), z3(u)), wherez2(u)
and z3(u) denote the altitude and gradient covariates given in Figure 3. An
exponential covariance function c(u, v) = σ^{2}exp(−ku−vk/α) is used for the
Gaussian process, whereσandαare positive parameters.

4.2.2 Shot noise Cox processes

Ashot noise Cox processXhas random intensity function Λ(u) = X

(c,γ)∈Φ

γk(c, u) (10)

where c ∈ R^{2}, γ > 0, Φ is a Poisson process on R^{2}×(0,∞), and k(c,·) is a
density for a two-dimensional continuous random variable (Møller, 2003). Note
that X is distributed as the superposition (i.e. union) of independent Poisson
processesX(c,γ) with intensity functionsγk(c,·), (c, γ)∈Φ, where we interpret
X(c,γ) as a cluster with centre c and mean number of pointsγ. Thus X is an
example of a Poisson cluster process (Bartlett, 1964), and provides a natural
model for seed setting mechanisms causing clustering, see e.g. Brix & Chadoeuf
(2002). Simple formulae for the intensity and pair correlation functions of a
shot noise Cox process are provided in Møller (2003).

Example 4.2. (Shot noise Cox process for minke whales) In Waagepetersen &

Schweder (2006), the positions of minke whales in Figure 1 are modelled as an independent thinning of a shot noise Cox process. Lettingp(u) denote the probability of observing a whale at locationu, the process of observed whales is a Cox process driven by Λ(u) =p(u)P

(c,γ)∈Φγk(c, u). The cluster centres
are assumed to form a stationary Poisson process with intensityκ, thec’s are
independent of the γ’s, and the γ’s are i.i.d. gamma random variables with
meanαand unit scale parameter. To handle edge effects,k(c,·) is the density
ofN2(c, ω^{2}I) restricted toc+ [−3ω,3ω]^{2}.

A particular simple case of a shot noise Cox process is aNeyman-Scott process X, where the centre points form a stationary Poisson process with intensityκ and the γ’s are all equal to a positive parameter α (Neyman & Scott, 1958).

If furthermorek(c,·) is a bivariate normal density with meancand covariance
matrixω^{2}I, thenXis a(modified) Thomas process(Thomas, 1949). A Neyman-
Scott process is stationary with intensityρ =ακ, and the Thomas process is

also isotropic with

g(r) = 1 + exp −r^{2}/(4ω^{2})

/(4πκω^{2}), r >0. (11)
Shot noise Cox process can be extended in various interesting ways by al-
lowing the kernelkto depend on a random band widthband replacingΦby a
Poisson or non-Poisson process model for the points (c, γ, b) (Møller & Torrisi,
2005). In this paper, we consider instead an extension which incorporates co-
variate information in a multiplicative way, i.e. an inhomogeneous Cox process
driven by

Λ(u) = exp z(u)β^{T} X

(c,γ)∈Φ

γk(c, u) (12)

(Waagepetersen, 2005). A nice feature is that the pair correlation function of Xis the same for (10) and (12), i.e. it does not depend on the parameterβ.

Example 4.3. (Inhomogeneous Thomas model for tropical rain forest trees) In addition to the log Gaussian Cox process model for the tropical trees (Exam- ple 4.1), we consider an inhomogeneous Thomas process of the form (12),

Λ(u) = α

2πω^{2}exp (β2z2(u) +β3z3(u))X

c∈Φ

exp −ku−ck^{2}/(2ω^{2})

where nowΦ denotes a stationary Poisson process with intensityκ. Then the intensity function is

ρ(u) =καexp (β2z2(u) +β3z3(u)) (13) while the pair correlation function is equal to (11).

### 5 Modelling the conditional intensity function

Gibbs point processes arose in statistical physics as models for interacting parti- cle systems. The intensity function for a Gibbs process is unknown; instead, the Papangelou conditional intensityλ(u,x) (Papangelou, 1974) becomes the appro- priate starting point for modelling. The definition and interpretation ofλ(u,x) are given in Section 5.2 in terms of the density of a finite point process and in Section 5.4 using a more technical account for infinite point processes. The den- sity of a Gibbs point process is specified in Section 5.3. Though the density has an unknown normalizing constant, likelihood inference based on MCMC meth-

ods is easier for parametric Gibbs point process models than for Cox processes;

see Section 7. While Cox processes provide flexible models for aggregation or clustering in a point pattern, Gibbs point processes provide flexible models for regularity or repulsion (Sections 5.3 and 10.3).

### 5.1 Finite point processes with a density

Throughout this section we assume that S is bounded and Xis a finite point process defined on S. Moreover, Yρ denotes a Poisson process on S with in- tensity measure µ and intensity function ρ. In particular, Y1 is theunit rate Poisson processon S with intensity ρ≡ 1. Before defining what is meant by the density ofX, we need the following useful Poisson expansion. IfF denotes any event of spatial point patterns contained inS, by (i)-(ii) in Section 4.1,

P(Yρ∈F) =

∞

X

n=0

e^{−µ(S)}
n!

Z

S^{n}

1[x∈F]ρ(x1)· · ·ρ(xn) dx1· · ·dxn (14)

wherex={x1, . . . , xn}.

By (14),Xhasdensityf with respect to Y1 if P(X∈F) = E [1[Y1∈F]f(Y1)]

=

∞

X

n=0

e^{−|S|}

n!

Z

S^{n}

1[x∈F]f(x) dx1· · ·dxn (15)

where|S|is the area ofS. Combining (3) and (15) it follows that

ρ^{(n)}(u1, . . . , un) = Ef(Y1∪ {u1, . . . , un}) (16)
for anyn∈Nand pairwise different points u1, . . . , un ∈S. Conversely, under
mild conditions,f can be expressed in terms of the product densitiesρ^{(n)}(Mac-
chi, 1975). Furthermore, conditional onn(X) =n withn≥1, thenpoints in
Xhave a symmetric joint density

fn(x1, . . . , xn)∝f({x1, . . . , xn}) (17)
onS^{n}.

Apart from the Poisson process and a few other simple models such as the mixed Poisson process (Grandell, 1997), the density is not expressible in closed

form. For the Poisson processYρ, (14) gives
f(x) = e^{|S|−µ(S)}

n

Y

i=1

ρ(xi). (18)

Thus, for a Cox process driven byΛ= (Λ(u))u∈S,

f(x) = E

"

exp

|S| − Z

S

Λ(u) du

Y

u∈X

Λ(u)

#

(19)

which in general is not expressible in closed form.

### 5.2 The conditional intensity

The usual conditional intensity of a one-dimensional point process does not ex-
tend to two-dimensional point processes because of the lack of a natural ordering
in R^{2}. Instead, the Papangelou conditional intensity becomes the appropriate
counterpart (Papangelou, 1974); a formal definition is given below.

Let the situation be as in Section 5.1, and suppose thatf ishereditary, i.e.

f(x)>0 andy⊂x ⇒ f(y)>0 (20) for finite point configurations x ⊂ S. This condition is usually assumed in practice.

Now, for locations u∈S and finite point configurationsx⊂S, the Papan- gelou conditional intensity is defined by

λ(u,x) =f(x∪ {u})/f(x\ {u}) (21) iff(x\ {u})>0, and λ(u,x) = 0 otherwise. The precise definition ofλ(u,x) whenu∈xis not that important, and (21) just covers this case for completeness.

Note thatλ(u,x) =λ(u,x\ {u}), and (20) implies thatf andλare in a one- to-one correspondence.

For a Poisson process, the Papangelou conditional intensity is simply the intensity: iff(x)>0 is given by (18), thenλ(u,x) =ρ(u) does not depend on x, again showing the absence of interaction in a Poisson process.

Combining (16) and (20)–(21),

ρ(u) = Eλ(u,X). (22)

Recall that the conditional probability P(A|x) of an event A given X = x satisfies P(A) = E[P(A|X)]. Thus due to the infinitesimal interpretation of ρ(u) du(Section 3.2), it follows from (22) thatλ(u,x) dumay be interpreted as the conditional probability that there is a point of the process in an infinitesimal region containing u and of area du given that the rest of the point process coincides withx.

Often a densityf is specified by anunnormalized densityh, i.e.f ∝hwhere his an hereditary function, for which the normalizing constant Eh(Y1) is well defined but unknown. However,

λ(u,x) =h(x∪ {u})/h(x\ {u})

does not depend on the normalizing constant. This is one reason why inference and simulation procedures are often based on the conditional intensity rather than the density of the point process.

In practically all cases of spatial point process models, an unnormalized densityhislocally stable, that is, there is a constantK such that

h(x∪ {u})≤Kh(x) (23)

for allu∈S and finitex⊂S. Local stability implies both thathis hereditary and integrable with respect the to unit rate Poisson process. Local stability also plays a fundamental role when studying stability properties of MCMC algorithms (Section 9.2).

### 5.3 Finite Gibbs point processes

Consider again a finite point processX defined on the bounded region S and with hereditary densityf. This is aGibbs point process(also called a canonical ensemble in statistical physics) if

logλ(u,x) = X

y⊆x

U(y∪ {u}) whenf(x)>0 (24)

where the function U(x) ∈ [−∞,∞) is defined for all non-empty finite point configurationsx⊂S, and we set log 0 =−∞. In statistical mechanical terms, U is a potential.

A large selection of Gibbs point process models are given in Van Lieshout (2000) and Møller & Waagepetersen (2003b). Usually, a log linear model is

considered, where the first order potential is either constant or depends on spatial covariates

U(u)≡U({u}) =z(u)β^{T}
and higher order potentials are of the form

U(x) =V(x)ψ_{n(x)}^{T} , n(x)≥2

where theψn are so-called interaction parameters. Thenλis parameterized by θ= (β, ψ2, ψ3, . . .) and is on log linear form

logλθ(u,x) = (t(x∪ {u})−t(x))θ^{T} (25)
where

t(x) =

X

u∈x

z(u), X

y⊆x:n(y)=2

V(y), X

y⊆x:n(y)=3

V(y), . . .

. (26) Combining (21) and (24), the Gibbs process has density

f(x)∝exp

X

∅6=y⊆x

U(y)

(27)

defining exp(−∞) = 0. Unless X is Poisson, i.e. when U(y) = 0 whenever n(y) ≥ 2, the normalizing constant of the density is unknown. Usually for models used in practice, U(y) ≤ 0 if n(y) ≥ 2, which implies local stability (and hence integrability). This means that the points in the process repel one other, so that realizations of the process tend to be more regular than for a Poisson process. Most Gibbs models are pairwise interaction processes, i.e.

U(y) = 0 whenevern(y)≥3, and typically the second order potential depends on distance only,U({u, v}) = U(ku−vk). Ahard core process with hard core r >0 hasU({u, v}) =−∞wheneverku−vk< r.

TheHammersley-Clifford theoremfor Markov random fields was modified to the case of spatial point processes by Ripley & Kelly (1977), stating that any hereditary density is of the form (27) and the following properties (I) and (II) are equivalent.

(I) U(x) = 0 whenever there exist two points{u, v} ⊆xsuch thatku−vk> R.

(II) Iff(x)>0 and u∈S\x, thenλ(u,x) =λ(u,x∩b(u, R)).

Hereb(u, R) is the closed disc with centreuand radius R. When (I) or (II) is satisfied, X is said to beMarkov withinteraction radius R, or more precisely, Markov with respect to the R-close neighbourhood relation. This definition and the Hammersley-Clifford theorem can be extended to an arbitrary sym- metric relation onS (Ripley & Kelly, 1977) or even a relation which depends on realizations of the point process (Baddeley & Møller, 1989). Markov point processes constitute a particular important subclass of Gibbs point processes, since the local Markov property (II) very much simplifies the computation of the Papangelou conditional intensity in relation to parameter estimation and simulation.

The property (I) implies a spatial Markov property. If B ⊂ S and ∂B = {u6∈B :b(u, R)∩B 6=∅}is its R-close neighbourhood, then the process XB

conditional onXB^{c}depends only onXB^{c}throughX∂B. The conditional process
XB|X∂B =x∂B is also Gibbs, with density

fB(xB|x∂B)∝exp

X

∅6=y⊆xB

U(y∪x∂B)

(28) where the normalizing constant depends onx∂B (the conditional density may be arbitrarily defined if U(y) = −∞ for some non-empty point configuration y⊆x∂B). The corresponding Papangelou conditional intensity is

λ(u,xB|x∂B) =λ(u,xB∪x∂B), u∈B. (29)
Example 5.1. (Overlap interaction model for Norwegian spruces) The condi-
tional intensity for a Norwegian spruce with a certain influence zone should
depend not only on the positions but also on the influence zones of the neigh-
bouring trees, see Figure 4. A tree with influence zone given by the discb(u, mu),
whereuis the spatial location of the tree andmuis the influence zone radius, is
treated as a point (u, mu) inR^{3}. Confining ourselves to a pairwise interaction
process, we define the pairwise potential by

U({(u, mu),(v, mv)}) =ψ|b(u, mu)∩b(v, mv)|, ψ≤0.

Hence, the strength of the repulsion between two trees (u, mu) and (v, mv) is given by ψ times the area of overlap between the influence zones of the two trees. We assume that the influence zone radii belong to a bounded interval

M = [a, b], whereaandb are estimated by the minimal and maximal observed influence zone radii. We divide M into six disjoint subintervals of equal size, and define the first order potential by

U({(u, mu)}) =β(mu) =βk ifmu falls in thekth subinterval

where βk is a real parameter. This enables modelling the varying numbers of trees in the six different size classes. However, the interpretation of the conditional intensity

λθ((u, mu),x) = exp

β(mu) +ψ X

(v,mv)∈x

|b(u, mu)∩b(v, mv)|

(30) is not straightforward — it is for instance not in general a monotone function of mu. On the other hand, for a fixed (u, mu), the conditional intensity will always decrease if the neighbouring influence zones increase.

Example 5.2. (Hierarchical model for ants nests) The hierarchical model in H¨ogmander & S¨arkk¨a (1999) for the positions of ants nests is based on so-called Strauss processes with hard coresand interaction range R = 45 (distances are measured in ft). Details follow below.

For distancest >0, define

V(t;r) =

−∞ ift≤r 1 ifr < t≤45 0 otherwise

where r≥0 denotes a hard core distance (or no hard core if r = 0). For the Messornests, the Strauss process with hard corerM is given by first and second order potentials

UM1({u}) =βM, UM2({u, v}) =ψMV(ku−vk;rM),

and no higher order interactions. The conditional intensity for a putative nest at a location uis thus zero if an existing nest occur within distance rM from u, and otherwise the log conditional density is given by the sum ofβM andψM

times the number of neighbouring nests within distance 45. Given the pattern xM ofMessor nests, the Cataglyphisnests are modelled as an inhomogeneous

Strauss process with one hard corerCM to theMessor nests and another hard corerC between theCataglyphisnests, i.e. using potentials

UC1({u}) =βC+ψCM

X

v∈xM

V(ku−vk;rCM), UC2({u, v}) =ψCV(ku−vk;rC).

Finally, the hard cores are estimated by the observed minimum interpoint dis- tances, which are biased maximum likelihood estimates.

Using positive hard coresrM andrC may be viewed as an ad hoc approach to obtain a model which is well-defined for all real values of the parameters βM, βC, ψM,ψCM, andψC, whereby both repulsive and attractive interaction within and between the two types of ants can be modelled. However, as noted by Møller (1994), the Strauss hard core process is a poor model for clustering due to the following ‘phase transition property’: for positive values of the interaction parameter, except for a narrow range of values, the distribution will either be concentrated on point patterns with one dense cluster of points or in ‘Poisson- like’ point patterns.

We also use maximum likelihood estimatesrM = 9.35 andrC= 2.45, but in contrast to H¨ogmander & S¨arkk¨a (1999) we find it more natural to consider a model with no hard core between the two types of ants nests, i.e. to letrCM = 0.

Figure 6 shows the support of the covariate function z2(u) = P

v∈xMV(ku− vk; 0) for theCataglyphismodel with rCM= 0.

### 5.4 Infinite Gibbs point processes

In general it is not possible to deal with densities of infinite point processes.

For example, a stationary Poisson process has a density with respect to another
stationary Poisson process if and only if their intensities are equal. However, the
Papangelou conditional intensityfor a point processXonR^{2}can be indirectly
defined as follows. If λ(u,x) is a non-negative function defined for locations
u∈R^{2} and locally finite point configurationsx⊂R^{2}such that

EX

u∈X

h(u,X\ {u}) = Z

E[λ(u,X)h(u,X)] du (31)

for all non-negative functions h, then λ is the Papangelou conditional inten- sity ofX. In fact two infinite point processes can share the same Papangelou conditional intensity; this phenomenon is known in statistical physics asphase transition.

0 200 400 600 800

0100200300400500600700

Figure 6: The white region is the set ofu∈W with distance less than 45 to a Messornest. The dots show the locations ofCataglyphisnests.

The integral formula (31) is called theGeorgii-Nguyen-Zessin formula(Geor- gii, 1976; Nguyen & Zessin, 1979), and this together with the Campbell theorem are basically the only known general formulae for spatial point processes. It is straightforward to verify (31) when Xis defined on a bounded region, so that it is a finite point process with Papangelou conditional intensity (21). Using induction we obtain theiterated GNZ-formula

E

6=

X

x1,...,xn∈X

h(x1, . . . , xn,X\ {x1, . . . , xn}) = Z

· · · Z

E[λ(x1,X)λ(x2,X∪ {x1})

· · ·λ(xn,X∪ {x1, . . . , xn−1})h(x1, . . . , xn,X)] dx1· · ·dxn (32) for non-negative functionsh. Combining (3) and (32), we see that

ρ^{(n)}(u1, . . . , un) = E[λ(u1,X)λ(u2,X∪ {u1})· · ·λ(un,X∪ {u1, . . . , u_{n−1}})].

(33) Notice that the iterated GNZ-formula (32) implies the Campbell theorem (4).

For instance, for a Cox process driven by Λ,

λ(u,X) = E [Λ(u)|X]. (34) However, this conditional expectation is usually unknown, and the GNZ-formula is more useful in connection with Gibbs point processes as described below.

The most common approach for defining aGibbs point processXonR^{2}is to
assume thatXsatisfies the spatial Markov property with respect to theR-close
neighbourhood relation, and has conditional densities of a similar form as in the
finite case. That is, for any bounded regionB⊂R^{2},XB|XB^{c} depends onXB^{c}

only through X∂B, and (28) specifies the conditional density. An equivalent
approach is to assume thatXhas a Papangelou conditional intensity, which in
accordance with (28) and (29) satisfiesλ(u,X) =λ(u,X∩b(u, R)), where for
finite point configurationsx⊂R^{2} and locationsu∈R^{2},

λ(u,x) = exp

X

y⊆x

U(y∪ {u})

ifu6∈x, λ(u,x) =λ(u,x\{u}) ifu∈x.

Unfortunately, (33) is not of much use here, and in general a closed form ex-
pression forρ^{(n)}is unknown whenXis Gibbs.

Questions of much interest in statistical physics are if a Gibbs process exists forλspecified by a given potentialU as above, and if the process is unique (i.e.

no phase transition) and stationary (even in that case it may not be unique);

see Ruelle (1969), Preston (1976), Georgii (1976), Nguyen & Zessin (1979) or the review in Møller & Waagepetersen (2003b). These questions are of less importance in spatial statistics, where the process is observed within a bounded window W and, in order to deal with edge effects, we may use the so-called border method. That is, we base inference onXW R|X∂W R, whereW R is the clipped observation window

W_{ R}={u∈W :b(u, R)⊂W}

and the Papangelou conditional intensity is given by λ(u,xW R|x∂W R) = λ(u,x) when XW = x is observed. We return to this issue in Sections 6.1.3 and 7.2.

### 6 Exploratory and diagnostic tools

It is often difficult to assess the properties of a spatial point pattern by eye. A realization of a homogeneous Poisson process may for example appear clustered due to points which happen to be close just by chance. This section explains how to explore the features of a spatial point pattern with the aim of suggesting an appropriate model, and how to check and critize a fitted model. The resid- uals described in Section 6.1 are useful to assess the adequacy of the specified (conditional) intensity function in relation to a given data set. The second order properties specified by the pair correlation function and the distribution of in- terpoint distances may be assessed using the more classical summary statistics in Section 6.2.

In this section, ˆρand ˆλdenote estimates of the intensity function and the Pa- pangelou conditional intensity, respectively. These estimates may be obtained by non-parametric or parametric methods. In the stationary case, or at least if ρis constant onS, a natural unbiased estimate is ˆρ=n/|W|. In the inhomoge- neous case, a non-parametric kernel estimate is

ˆ ρ(u) =

n

X

i=1

k(u−xi)/

Z

W

k(v−u) dv (35)

where k is a kernel with finite band width, and where the denominator is an edge correction factor ensuring thatR

Wρ(u) duˆ is an unbiased estimate ofµ(W) (Diggle, 1985). If the intensity or conditional intensity is specified by a para- metric model,ρ=ρθ or λ=λθ, and θis estimated by ˆθ(x) (Sections 7–8), we let ˆρ=ρθ(x)ˆ or ˆλ=λθ(x)ˆ .

### 6.1 Residuals

For a Gibbs point process with log Papangelou conditional intensity (24), the first order potential corresponds to the linear predictor of a generalised lin- ear model (GLM), while the higher order potentials are roughly analogous to the distribution of the errors in a GLM. Recently, Baddeley, Turner, Møller &

Hazelton (2005) developed a residual analysis for spatial point processes based on the GNZ-formula (31) and guided by the analogy with residual analysis for (non-spatial) GLM’s. For a Cox process, the Papangelou conditional intensity (34) is usually not expressible in closed form, while the intensity function may be tractable. In such cases, Waagepetersen (2005) suggested residuals be de-

fined using instead the intensity function. Whether we base residuals on the conditional intensity or the intensity, the two approaches are very similar.

6.1.1 Definition of innovations and residuals

For ease of exposition we assume first that the point processX is defined on the observation windowW; the case whereXextends outsideW is considered in Section 6.1.3.

For non-negative functionsh(u,x), define theh-weighted innovationby Ih(B) = X

u∈XB

h(u,X\ {u})− Z

B

λ(u,X)h(u,X) du, B ⊆W. (36)

We will allow infinite values of h(u,x) if u ∈ x, in which case we define λ(u,x)h(u,x) = 0 if λ(u,x) = 0. Baddeley et al. (2005) study in particular theraw, Pearson, and inverse-λinnovations given by h(u,x) = 1,1/p

λ(u,x), 1/λ(u,x), respectively. Note that Ih is a signed measure, where we may inter- pret ∆I(u) =h(u,X\ {u}) as the innovation increment (‘error’) attached to a pointuinX, and dI(u) =−λ(u,X)h(u,X)duas the innovation increment at- tached to a background locationu∈W. Assuming that the sum or equivalently the integral in (36) has finite mean, the GNZ-formula (31) gives

EIh(B) = 0. (37)

Theh-weighted residual is defined by Rˆh(B) = X

u∈xB

ˆh(u,x\ {u})− Z

B

λ(u,ˆ x)ˆh(u,x) du, B ⊆W, (38)

where, as the functionhmay depend on the model, ˆhdenotes an estimate. This also is a signed measure, and we hope that the mean of the residual measure is approximately zero. The raw, Pearson, and inverse-λresiduals are

R(B) =n(x)− Z

B

λ(u,ˆ x) du,

R1/√_{ˆ}

λ(B) = X

u∈xB

1/

qλ(u,ˆ x)− Z

B

qλ(u,ˆ x) du,

R_{1/}ˆλ(B) = X

u∈xB

1/λ(u,ˆ x)− Z

B

1[ˆλ(u,x)>0] du.

In order that the Pearson and inverse-λ residuals be well defined, we require that ˆλ(u,x)>0 for allu∈x. Properties of these innovations and residuals are analyzed in Baddeley, Møller and Pakes (2006).

Similarly, we define innovations and residuals based onρ, where in all expres- sions above we replaceλand ˆλbyρand ˆρ, respectively, andh(u,x) and ˆh(u,x) byh(u) and ˆh(u), respectively. Here it is required thatR

Wh(u)ρ(u) du <∞, so that (37) also holds in this case.

6.1.2 Diagnostic plots

Baddeleyet al. (2005) suggest variousdiagnostic plotsfor spatial trend, depen- dence of covariates, interaction between points, and other effects. In particular, the plots can check for the presence of such features when the fitted model does not include them. The plots are briefly described below in the case of residuals based onλ; if we instead consider residuals based onρ, we use the same substi- tutions as in the preceding paragraph. Figures 7 and 8 show specific examples of the plots in the case of theCataglyphis nests model (Example 5.2) fitted in Example 8.2 and based on raw residuals (h≡1). The plots are corrected for edge effects, cf. Section 6.1.3.

Themark plotis a pixel image with greyscale proportional to ˆλ(u,x)ˆh(u,x) and a circle centred at each pointu∈xwith radius proportional to the residual mass ˆh(u,x\ {u}). The plot may sometimes identify ‘extreme points’. For example, for Pearson residuals and a fitted model of correct form, large/small circles and dark/light greyvalues should correspond to low/high values of the conditional intensity, and in regions of the same greylevel the circles should be uniformly distributed. The upper left plot in Figure 7 is a mark plot for the raw residuals obtained from the model fitted to theCataglyphisnests in Example 8.2.

In this case, the circles are of the same radii and just show the locations of the nests. In the region of the large cluster of circles one could perhaps have expected larger values (more light grey scales) of the fitted conditional intensity.

Thesmoothed residual fieldat locationu∈W is s(u,x) =

Pn

1k(u−xi)ˆh(xi,x\ {xi})−R

Wk(u−v)ˆλ(u,x)ˆh(v,x) dv R

Wk(u−v) dv (39)

wherekis a kernel and the denominator is an edge correction factor. For exam- ple, for raw residuals, the numerator of (39) has mean R

Wk(u−v)E[λ(v,X)− ˆλ(v,X)] dv, so positive/negative values of s suggest that the fitted model un-

der/overestimates the intensity function. The smoothed residual field may be presented as a greyscale image and a contour plot. For example, the lower right plot in Figure 7 suggests some underestimation of the conditional intensity at the middle of the plot and overestimation in the top part of the plot.

For a given covariate z : W 7→R and numbers t, define W(t) = {u∈W : z(u) ≤ t}. A plot of the ‘cumulative residual function’ A(t) = Rˆh(W(t)) is called alurking variable plot, since it may detect ifz should be included in the model. If the fitted model is correct, we expect A(t) ≈ 0. The upper right and lower left plots in Figure 7 show lurking variable plots for the covariates given by the y and xspatial coordinates, respectively. The upper right plot indicates (in accordance with the lower right plot) a decreasing trend in they direction, whereas there is no indication of trend in thexdirection. The possible defects of the model indicated by the right plots in Figure 7 might be related to inhomogeneity; the observation window consists of a ‘field’ and a ‘scrub’ part divided by a boundary which runs roughly along the diagonal from the lower left to the upper right corner (Harkness & Isham, 1983). Including covariates given by an indicator for the field and the spatial y-coordinate improved somewhat the appearance of the diagnostic plots.

Baddeleyet al. (2005) also consider aQ-Q plotcomparing empirical quan-
tiles ofs(u,x) with corresponding expected empirical quantiles estimated from
s(u,x^{(1)}), . . . , s(u,x^{(n)}), where x^{(1)}, . . . ,x^{(n)} are simulations from the fitted
model. This is done using a grid of fixed locationsuj ∈W, j = 1, . . . , J. For
eachk = 0, . . . , n, where x^{(0)} =x is the data, we sort s^{(k)}_{j} =s(uj,x^{(k)}), j =
1, . . . , Jto obtain the order statisticss^{(k)}_{[1]} ≤. . .≤s^{(k)}_{[J]}. We then plots^{(0)}_{[j]} versus
the estimated expected empirical quantilePn

k=1s^{(k)}_{[j]}/n for j = 1, . . . , J. The
Q-Q plot in Figure 8 shows some deviations between the observed and estimated
quantiles but each observed order statistic fall within the 95% intervals obtained
from corresponding simulated order statistics.

6.1.3 Edge effects

Substantial bias and other artifacts in the diagnostic plots for residuals based on
λmay occur if edge effects are ignored. We therefore use the border method as
follows (see also Baddeleyet al., 2006b). Suppose the fitted model is Gibbs with
interaction radiusR (Sections 5.3-5.4). For locationsuin W\W_{ R} =∂W_{ R},
λ(u,x) may depend on points inxwhich are outside the observation windowW.
Since the Papangelou conditional intensity (29) withB=W Rdoes not depend

0 200 400 600 800 x coordinate

−2−1012

cumulative sum of raw residuals 0200400600 y coordinate

6 4 2 0 −2 −4

cumulative sum of raw residuals

Figure 7: Plots forCataglyphisnests based on raw residuals: mark plot (upper left), lurking variable plots for covariates given byy andxcoordinates (upper right, lower left), and smoothed residual field (lower right). Dark grey scales correspond to small values.

on points outside the observation window, we condition onX∂W R = x∂W R

and plot residuals only foru∈W_{ R}. See e.g. the upper left plot in Figure 7.

For residuals based onρinstead, we have no edge effects, so no adjustment of the diagnostic tools in Section 6.1.2 is needed.

### 6.2 Summary statistics

This section considers the more classical summary statistics such as Ripley’sK- function and the nearest-neighbour functionG. See also Baddeley, Møller and Waagepetersen (2006) who develop residual versions of such summary statistics.

−6e−05 −4e−05 −2e−05 0e+00 2e−05 4e−05 6e−05

−4e−05−2e−050e+002e−054e−056e−05

Mean quantile of simulations

data quantile

Residuals: raw

Figure 8: Q-Q plot forCataglyphisnests based on smoothed raw residual field.

The dotted lines show the 2.5% and 97.5% percentiles for the simulated order statistics

6.2.1 Second order summary statistics

Second order properties are described by the pair correlation functiong, where it is convenient if g(u, v) only depends on the distance ku−vk or at least the difference u−v (note that g(u, v) is symmetric). Kernel estimation of g is discussed in Stoyan & Stoyan (2000). Alternatively, if g(u, v) = g(u−v) is translation invariant, one may consider the inhomogeneous reduced second moment measure(Baddeleyet al., 2000)

K(B) = Z

B

g(u) du, B⊆R^{2}.

More generally, ifg is not assumed to exist or to be translation invariant, we may define

K(B) = 1

|A|E X

u∈XA

X

v∈X\{u}

1[u−v∈B]

ρ(u)ρ(v) (40)

provided thatXissecond order reweighted stationarywhich means that the right
hand side of (40) does not depend on the choice ofA⊂R^{2}, where 0<|A|<∞.

Note thatK is invariant under independent thinning.

The (inhomogeneous) K-function is defined by K(r) = K(b(0, r)), r > 0.

Clearly, if g(u, v) = g(ku−vk), then K is determined by K, and K(r) = 2πRr

0 sg(s) ds, so that g and K are in a one-to-one correspondence. In the
case of a stationary point process, it follows from (40) that ρK(r) has the
interpretation as the expected number of further points within distancer from
a typical point inX, andρ^{2}K(r)/2 is the expected number of (unordered) pairs
of distinct points not more than distancerapart and with at least one point in
a set of unit area (Ripley, 1976). A formal definition of ‘typical point’ is given
in terms of Palm measures, see e.g. Møller & Waagepetersen (2003b). For a
Poisson process,K(r) =πr^{2}.

In our experience, non-parametric estimation ofKis more reliable than that ofg, since the latter involves kernel estimation, which is sensitive to the choice of the band width. Various edge corrections have been suggested, the simplest and most widely applicable being

K(r) =ˆ

6=

X

u,v∈x

1[ku−vk ≤r]

ˆ

ρ(u)ˆρ(v)|W∩W_{u−v}| (41)
whereWu isW translated by u, and ˆρis an estimate of the intensity function.

One possibility is the non-parametric estimate ofρgiven in (35) but the resulting estimate ˆK(r) is very sensitive to the choice of kernel band width. In general we prefer to use a parametric estimate of the intensity function.

An estimate of theK-function for the tropical rain forest trees obtained with a parametric estimate of the intensity function (see Example 8.1) is shown in Figure 9. The plot also shows theoretical K-functions for fitted log Gaussian Cox, Thomas, and Poisson processes, where all three processes share the same intensity function (details are given later in Example 8.3). The trees seem to form a clustered point pattern since the estimatedK-function is markedly larger than the theoreticalK-function for a Poisson process.

One often considers the L-functionL(r) =p

K(r)/π, which at least for a stationary Poisson process is a variance stabilizing transformation when K is estimated by non-parametric methods (Besag, 1977). Moreover, for a Poisson process, L(r) =r. In general, at least for small distances, L(r)> r indicates aggregation andL(r)< r indicates regularity. Usually when a model is fitted, L(r) =ˆ

qK(r)/πˆ or ˆL(r)−r is plotted together with the average and 2.5%

and 97.5% quantiles based on simulated ˆL-functions under the fitted model; we

0 20 40 60 80 100

010000200003000040000

r

K(r)

Estimate Poisson Thomas LGCP

Figure 9: Estimated K-function for tropical rain forest trees and theoretical K-functions for fitted Thomas, log Gaussian Cox, and Poisson processes.

refer to these bounds as 95%envelopes. Examples are given in the right plots of Figures 11 and 12.

Estimation of third-order properties and of directional properties (so-called directionalK-functions) is discussed in Stoyan & Stoyan (1995), Møller et al.

(1998), Schladitz & Baddeley (2000), and Guan, Sherman & Calvin (2006).

6.2.2 Interpoint distances

In order to interpret the following summary statistics based on interpoint dis- tances, we assume stationarity ofX. Theempty space functionF is the distri- bution function of the distance from an arbitrary location to the nearest point inX,

F(r) = P(X∩b(0, r)6=∅), r >0.

Thenearest-neighbour functionis defined by G(r) = 1

ρ|W|E X

u∈X∩W

1[(X\ {u})∩b(u, r)], r >0,

which has the interpretation as the cumulative distribution function for the
distance from a ‘typical’ point inXto its nearest-neighbour point inX. Thus,
for small distances,G(r) andρK(r) are closely related. For a stationary Poisson
process,F(r) =G(r) = 1−exp(−πr^{2}). In general, at least for small distances,
F(r)> G(r) indicates aggregation andF(r)< G(r) indicates regularity. Van
Lieshout & Baddeley (1996) study the nice properties of theJ-functiondefined
byJ(r) = (1−G(r))/(1−F(r)) for F(r)<1.

Non-parametric estimation ofFandGaccounting for edge effects is straight- forward using border methods, see e.g. Møller & Waagepetersen (2003b). An estimate ofJ is obtained by plugging in the estimates ofF andGin the expres- sion forJ. We combine the estimates to obtain an estimate ofJ. Estimates of F,G, andJ for the positions of Norwegian spruces shown in Figure 10 provide evidence of repulsion.

r

F(r)

0 1 2 3 4

0.00.20.40.60.81.0

r

G(r)

0 1 2 3 4

0.00.20.40.60.81.0

r

J(r)

0 1 2 3

12345

Figure 10: Left to right: estimated F, G, and J-functions for the Norwegian spruces (solid lines) and 95% envelopes calculated from simulations of a homo- geneous Poisson process (dashed lines) with expected number of points equal to the observed number of points. The long-dashed curves show the theoretical values ofF,G, andJ for a Poisson process.

### 7 Likelihood-based inference and MCMC meth- ods

Computation of the likelihood function is usually easy for Poisson process mod- els (Section 7.1), while the likelihood contains an unknown normalizing constant for Gibbs point process models, and is given in terms of a complicated integral for Cox process models. Using MCMC methods, it is now becoming quite feasi- ble to compute accurate approximations of the likelihood function for Gibbs and Cox process models (Sections 7.2 and 7.3). However, the computations may be

time consuming and standard software is yet not available. Quick non-likelihood approaches to inference are reviewed in Section 8.

### 7.1 Poisson process models

For a Poisson process with a parameterized intensity functionρθ, the log likeli- hood function is

l(θ) =X

u∈x

logρθ(u)− Z

W

ρθ(u) du, (42)

cf. (18), where in general numerical integration is needed to compute the in- tegral. A clever implementation for finding the maximum likelihood estimate (MLE) numerically, based on software for generalized linear models (Berman and Turner, 1992), is available inspatstat when the intensity function is of the log linear form (7).

Rathbun & Cressie (1994) study increasing domain asymptotics for inhomo- geneous Poisson point processes and provide fairly weak conditions for asymp- totic normality of the MLE in the case of a log linear intensity function. Waa- gepetersen (2005) instead suggests asymptotics for a fixed observation window when the intercept in the log linear intensity function tends to infinity, and the only condition for asymptotic normality of the MLE of the remaining parame- ters is positive definiteness of the observed information matrix. Inference for a log linear Poisson process model is exemplified in Example 8.1.

### 7.2 Gibbs point process models

We restrict attention to parametric models for Gibbs point processesX as in Sections 5.3–5.4, assuming that the interaction radiusR is finite and the con- ditional intensity is on the log linear form (25) (no matter whetherXis finite or infinite). We assume to begin with thatR is known.

First, suppose that the observation windowW coincides withS. The density is then on exponential family form

fθ(x) = exp(t(x)θ^{T})/cθ

wheretis given by (26) andcθ is the unknown normalizing constant. The score function and observed information are

u(θ) =t(x)−Eθt(X), j(θ) = Varθt(X),